Method of calibrating spectral apparatus and method of producing calibrated spectral apparatus

ABSTRACT

Creating a model calibrating spectral apparatus having an optical system that converts light to be measured into a spectrum, and a light-receiving sensor including a plurality of sensors that outputs signals, the sensors include sensors that output signals indicating respective energy amounts of a plurality of wavelength components. The model shows where a linear function of an indicator indicating a mechanical error in the spectral apparatus, expresses deviation of an indicator indicating spectral sensitivity of the sensor from the indicator indicating the reference spectral sensitivity of the sensor. The method comprises: a) acquiring reference spectral sensitivity; b) acquiring an indicator indicating the reference spectral sensitivity of the sensor acquired at a); and c) creating the model where the linear function of the mechanical error indicator expresses deviation of the spectral sensitivity indicator from the indicator indicating the reference spectral sensitivity of the sensor, acquired at b).

RELATED APPLICATION

This application is a Continuation of U.S. application Ser. No.15/746,216 filed Jan. 19, 2018 which is a U.S. National StageApplication under 35 USC § 371 of International application No.PCT/JP2016/069999 filed Jul. 6, 2016, which claims priority of Japanesepatent application no. 2015-149428 filed Jul. 29, 2015, the entirecontent of all of which are hereby incorporated by reference.

TECHNICAL FIELD

The present invention relates to calibration of a spectral apparatus.

BACKGROUND ART

A spectral apparatus including a polychromator, includes sensors thatoutput signals indicating the respective energy amounts of a pluralityof wavelength components. Spectral characteristics, such as spectralreflectance, spectral radiance, and spectral transmittance, are acquiredfrom the plurality of signals output by the plurality of sensors.

The spectral sensitivity of each of the sensors varies depending on, forexample, the arrangement, shape, and size of a light-receiving sensor,and varies depending on, for example, the arrangement, shape, and sizeof a slit plate or a diffraction grating included in the polychromator.Therefore, the spectral sensitivity of such a sensor, fundamental in thespectral characteristics, must be calibrated per spectral apparatus.

A wavelength calibration method of a spectral apparatus described inPatent Literature 1, includes: giving a difference from the referencevalue of the center wavelength in spectral sensitivity by a linearfunction of pixel number (paragraph 0026); giving a ratio to thereference value of the full width at half maximum in the spectralsensitivity by a linear function of pixel number (paragraph 0028);creating a corrected spectral sensitivity table with the centerwavelength in the spectral sensitivity and the ratio to the referencevalue of the full width at half maximum in the spectral sensitivity(paragraph 0029), and determining coefficients included in each of thelinear functions of pixel number such that calculated relative outputcalculated with the corrected spectral sensitivity table andemission-line wavelengths and measured relative output are valuesclosest to each other (paragraph 0033).

CITATION LIST Patent Literature

Patent Literature 1: JP 4660694 B2

SUMMARY OF INVENTION Technical Problem

The wavelength calibration method of a spectral apparatus described inPatent Literature 1, includes: giving a difference from the referencevalue of the center wavelength in spectral sensitivity by a linearfunction of pixel number; and giving a ratio to the reference value ofthe full width at half maximum in the spectral sensitivity by a linearfunction of pixel number. However, the difference may be considerablydifferent from the linear function of pixel number or the ratio may beconsiderably different from the linear function of pixel number, in apractical spectral apparatus. Giving the difference by the linearfunction of pixel number and giving the ratio by the linear function ofpixel number each are not favorable approximation, and thus the spectralsensitivity is not acquired precisely in these cases. Giving thedifference by a higher order function being a quadratic function ofpixel number or more and giving the ratio by a higher order functionbeing a quadratic function of pixel number or more, are considered, butcorrection of coefficients with the higher order functions greatlychanges calculated relative output, and thus the coefficients with whichthe calculated relative output and measured relative output are valuesclosest to each other, are not appropriately acquired and the spectralsensitivity is not appropriately acquired.

The invention below has been made in order to solve the problem. Anobject of the invention below is to acquire spectral sensitivityprecisely and appropriately.

Solution to Problem

A spectral apparatus to be calibrated, includes an optical system and alight-receiving sensor. The optical system converts light to be measuredinto a spectrum. The light-receiving sensor includes a plurality ofsensors that outputs a plurality of signals, the plurality of sensorsincluding sensors that outputs signals indicating the respective energyamounts of a plurality of wavelength components included in thespectrum.

For each of the plurality of sensors, the reference spectral sensitivityof the sensor is acquired, and an indicator indicating the referencespectral sensitivity of the sensor that has been acquired, is acquired.A model in which a linear function of an indicator indicating amechanical error in the spectral apparatus, expresses the deviation ofan indicator indicating the spectral sensitivity of the sensor from theindicator indicating the reference spectral sensitivity of the sensor,that has been acquired, is created. The indicator indicating themechanical error in the spectral apparatus, is acquired to adapt thespectral sensitivity of the sensor indicated by the indicator indicatingthe spectral sensitivity of the sensor, to the signal output by thesensor. For each of the plurality of sensors, the deviation of theindicator indicating the spectral sensitivity of the sensor from theindicator indicating the reference spectral sensitivity of the sensor,is acquired with the model that has been created and the indicatorindicating the mechanical error in the spectral apparatus, which hasbeen acquired. The spectral sensitivity of the sensor is acquired withthe reference spectral sensitivity of the sensor and the deviation ofthe indicator indicating the spectral sensitivity of the sensor from theindicator indicating the reference spectral sensitivity of the sensor,which have been acquired.

Advantageous Effects of Invention

Since the linear function of the indicator indicating the mechanicalerror in the spectral apparatus expresses the indicator indicating thespectral sensitivity of the sensor, the adaptability of a spectralsensitivity set to a signal set does not greatly vary even in a casewhere the indicator indicating the mechanical error in the spectralapparatus, varies. Therefore, the indicator indicating the mechanicalerror in the spectral apparatus, is appropriately acquired, and thespectral sensitivity of the sensor is acquired precisely andappropriately.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a schematic view of a spectral apparatus and a calibratingdevice.

FIG. 2 is a schematic view of a spectral unit.

FIG. 3 is a schematic view of a linear array sensor.

FIG. 4 is a block diagram of a signal processing mechanism.

FIG. 5 is a graph exemplarily illustrating the spectral sensitivity ofeach sensor and an emission-line spectrum.

FIG. 6 is a schematic graphical representation of the relationshipbetween reference spectral sensitivity and spectral sensitivity aftercalibration.

FIG. 7 is a graph of the relationship between the center wavelength inreference spectral sensitivity and the deviated amount of the centerwavelength.

FIG. 8 is a graph of the relationship between the center wavelength inthe reference spectral sensitivity and the ratio in full width at halfmaximum.

FIG. 9 is a flowchart of a procedure of creating a third model.

FIG. 10 is a graph of deviated amount of the center wavelength in thechannel of each sensor, the deviated amount being caused by a mechanicalerror a1.

FIG. 11 is a graph of deviated amount of the full width at half maximumin the channel of each sensor, the deviated amount being caused by themechanical error a1.

FIG. 12 is a graph of deviated amount of the center wavelength in thechannel of each sensor, the deviated amount being caused by a mechanicalerror a2.

FIG. 13 is a graph of deviated amount of the full width at half maximumin the channel of each sensor, the deviated amount being caused by themechanical error a2.

FIG. 14 is a graph of deviated amount of the center wavelength in thechannel of each sensor, the deviated amount being caused by a mechanicalerror a3.

FIG. 15 is a graph of deviated amount of the full width at half maximumin the channel of each sensor, the deviated amount being caused by themechanical error a3.

FIG. 16 is a graph of deviated amount of the center wavelength in thechannel of each sensor, the deviated amount being caused by a mechanicalerror a4.

FIG. 17 is a graph of deviated amount of the full width at half maximumin the channel of each sensor, the deviated amount being caused by themechanical error a4.

FIG. 18 is a graph of deviated amount of the center wavelength in thechannel of each sensor, the deviated amount being caused by a mechanicalerror a5.

FIG. 19 is a graph of deviated amount of the full width at half maximumin the channel of each sensor, the deviated amount being caused by themechanical error a5.

FIG. 20 is a flowchart of a procedure of calibrating the spectralapparatus.

FIG. 21 is a graph exemplarily illustrating the spectral sensitivity ofeach sensor and an emission-line spectrum.

FIG. 22 is a graph of the relationship between the channels of thesensors and measured signals in emission-line light source measurement.

FIG. 23 is a flowchart of a procedure of producing a calibrated spectralapparatus.

DESCRIPTION OF EMBODIMENTS

1. Spectral Apparatus

The schematic view of FIG. 1 illustrates a spectral apparatus and acalibrating device. FIG. 1 illustrates a section of the spectralapparatus.

As illustrated in FIG. 1, the spectral apparatus 100 includes a spectralunit 104 and a signal processing mechanism 105.

In a case where the spectral apparatus 100 performs measurement, thespectral unit 104 receives light to be measured 108 and outputs a firstsignal, a second signal, . . . , and a fortieth signal indicating theenergy amounts of a first wavelength component, a second wavelengthcomponent, . . . , and a fortieth wavelength component included in thelight to be measured 108 that has been received, respectively. Thesignal processing mechanism 105 acquires spectral characteristics withthe signals of the first signal, the second signal, . . . , and thefortieth signal.

In a case where the spectral apparatus 100 is a spectrophotometer, whenan object to be measured is irradiated with light, the light reflectedby the object to be measured is to be the light to be measured 108 and,for example, spectral reflectance is acquired as a spectralcharacteristic. In a case where the spectral apparatus 100 is aspectroradiometer, light emitted by a light source to be measured, is tobe the light to be measured 108 and, for example, spectral radiance isacquired as a spectral characteristic. In a case where the object thathas been measured is irradiated with light, the light transmittedthrough the object to be measured, may be the light to be measured 108and, for example, spectral transmittance may be acquired as a spectralcharacteristic. A colorimetric value may be acquired as a spectralcharacteristic. The colorimetric value is expressed by, for example, theMunsell color system, the L*a*b* color system, the L*C*h color system,the Hunter Lab color system, or the XYZ color system.

2. Spectral Unit

The schematic view of FIG. 2 is a perspective view of the spectral unit104.

The spectral unit 104 includes an optical system 111 and a linear arraysensor 112, as illustrated in FIGS. 1 and 2. The optical system 111 is apolychromator, and includes a slit plate 115 and a concave diffractiongrating 116.

The schematic view of FIG. 3 is a plan view of the linear array sensor.

The linear array sensor 112 includes sensors 119-1, 119-2, . . . , and119-40, as illustrated in FIG. 3.

In a case where the spectral apparatus 100 performs measurement, thelight to be measured 108 is guided to a rectangular slit 122 formedthrough the slit plate 115. The light to be measured 108 guided to theslit 122 passes through the slit 122. The light to be measured 108 thathas passed through the slit 122, travels from the slit 122 to thediffraction surface 124 of the concave diffraction grating 116, and thenis reflected on the diffraction surface 124. The light to be measured108 is reflected on the diffraction surface 124, so as to be convertedinto a spectrum. The light to be measured 108 converted into thespectrum, travels from the diffraction surface 124 of the concavediffraction grating 116 to the light-receiving surface 125 of the lineararray sensor 112, and then is image-formed on the light-receivingsurface 125, so as to be received by the linear array sensor 112. Thesensors 119-1, 119-2, . . . , and 119-40 are linearly arranged in adirection 128 on the light-receiving surface 125. The slit plate 115,the concave diffraction grating 116, and the linear array sensor 112 arearranged so that the light image-formed on the light-receiving surface125 varies in wavelength in accordance with position in the direction128. Therefore, in a case where the linear array sensor 112 receives thelight to be measured 108 converted into the spectrum, the sensors 119-1,119-2, . . . , and 119-40 output the first signal, the second signal, .. . , and the fortieth signal indicating the energy amounts of the firstwavelength component, the second wavelength component, . . . , and thefortieth wavelength component different from each other, respectively.The first signal, the second signal, . . . , and the fortieth signalthat have been output, are input into the signal processing mechanism105. The signal processing mechanism 105 acquires spectralcharacteristics with the first signal, the second signal, . . . , andthe fortieth signal that have been input.

The optical system 111 has an optical axis 134 leading from the slit 122to the diffraction surface 124 and an optical axis 135 leading from thediffraction surface 124 to the light-receiving surface 125.

The optical system 111 may be replaced with a different type of opticalsystem. For example, the concave diffraction grating 116 may be replacedwith a plane diffraction grating and a concave mirror. The slit plate115 and the concave diffraction grating 116 may be replaced with a slitplate having a circular slit formed, a cylindrical lens, and a linearvariable filter. Light to be measured that has passed through thecircular slit, passes through the cylindrical lens. The pass of thelight to be measured, through the cylindrical lens converts thesectional shape of the light to be measured from a circular shape into alinear shape. The light to be measured that has passed through thecylindrical lens, passes through the linear variable filter. The lightto be measured passes through the linear variable filter, so as to beconverted into a spectrum.

The linear array sensor 112 may be replaced with a different type oflight-receiving sensor. For example, the linear array sensor 112 may bereplaced with a linear array sensor including not more than 39 sensorsor not less than 41 sensors. Depending on an optical system, the lineararray sensor 112 may be replaced with an area sensor.

3. Signal Processing Mechanism

The block diagram of FIG. 4 illustrates the signal processing mechanism.

The signal processing mechanism 105 includes an A/D conversion mechanism138 and a computing mechanism 139, as illustrated in FIG. 4.

In a case where the first signal, the second signal, . . . , and thefortieth signal are input into the signal processing mechanism 105, thefirst signal, the second signal, . . . , and the fortieth signal areinput into the A/D conversion mechanism 138. The first signal, thesecond signal, . . . , and the fortieth signal input into the A/Dconversion mechanism 138, are analog-to-digital-converted into a firstsignal value, a second signal value, . . . , and a fortieth signalvalue, respectively. The first signal value, the second signal value, .. . , and the fortieth signal value are input into the computingmechanism 139. The computing mechanism 139 acquires the spectralcharacteristics with the first signal value, the second signal value, .. . , and the fortieth signal value that have been input, and thespectral sensitivity 140-1 of the sensor 119-1, the spectral sensitivity140-2 of the sensor 119-2, . . . , and the spectral sensitivity 140-40of the sensor 119-40 stored in the computing mechanism 139. Instead ofthe spectral sensitivity 140-1, 140-2, . . . , and 140-40, informationderived from the spectral sensitivity 140-1, 140-2, . . . , and 140-40,information being necessary for acquiring the spectral characteristics,may be stored in the computing mechanism 139.

The computing mechanism 139 is an embedded computer, and operates inaccordance with an installed program. The entirety or part of processingto be performed by the computing mechanism 139, may be performed by anelectronic circuit accompanying no program. The entirety or part ofprocessing to be performed by the computing mechanism 139, may bemanually performed.

4. Exemplary Spectral Sensitivity of Each Sensor

The graph of FIG. 5 exemplarily illustrates the spectral sensitivity ofeach of the sensors and the emission-line spectrum of light to bemeasured for emission-line calibration.

The center wavelength in the spectral sensitivity 141-1 of the sensor119-1, the center wavelength in the spectral sensitivity 141-2 of thesensor 119-2, . . . , and the center wavelength in the spectralsensitivity 141-40 of the sensor 119-40 are different from each other,and are approximately 352 nm, approximately 363 nm, . . . , andapproximately 740 nm, respectively, as illustrated in FIG. 5. With thisarrangement, the sensors 119-1, 119-2, . . . , and 119-40 output thefirst signal, the second signal, . . . , and the fortieth signalindicating the energy amounts of the first wavelength component, thesecond wavelength component, . . . , and the fortieth wavelengthcomponent different from each other, respectively.

5. Necessity of Calibration of Spectral Apparatus

For example, the arrangements, shapes, and sizes of the slit plate 115,the concave diffraction grating 116, and the linear array sensor 112,causes the spectral sensitivity 141-1, 141-2, . . . , and 141-40 tovary. Therefore, in order to acquire the spectral characteristicsprecisely, the spectral sensitivity 140-1, 140-2, . . . , and 140-40stored in the computing mechanism 139 must be changed in accordancewith, for example, the arrangements, shapes, and sizes of the slit plate115, the concave diffraction grating 116, and the linear array sensor,so as to be close to the real spectral sensitivity 141-1, 141-2, . . . ,and 141-40, respectively. Causing the spectral sensitivity 140-1, 140-2,. . . , and 140-40 stored in the computing mechanism 139 to be close tothe real spectral sensitivity 141-1, 141-2, . . . , and 141-40,respectively, is referred to as calibration of the spectral apparatus100.

6. Calibrating Device

The calibrating device 142 includes an HgCd lamp 143 and a controlcomputing mechanism 144, as illustrated in FIG. 1.

In a case where the calibration of the spectral apparatus 100 isperformed with the calibrating device 142, the control computingmechanism 144 causes the HgCd lamp 143 to emit the light to be measuredfor emission-line calibration as the light to be measured 108. Thespectral apparatus 100 measures the light to be measured foremission-line calibration that has been emitted. In a case where thespectral apparatus 100 measures the light to be measured foremission-line calibration, the first signal value, the second signalvalue, . . . , and the fortieth signal value are input into thecomputing mechanism 139. The computing mechanism 139 transfers the firstsignal value, the second signal value, . . . , and the fortieth signalvalue that have been input, to the control computing mechanism 144. Thecontrol computing mechanism 144 acquires the spectral sensitivity of thesensor 119-1, the spectral sensitivity of the sensor 119-2, . . . , andthe spectral sensitivity of the sensor 119-40 with the first signalvalue, the second signal value, . . . , and the fortieth signal valuethat has been transferred, respectively. The spectral sensitivity of thesensor 119-1, the spectral sensitivity of the sensor 119-2, . . . , andthe spectral sensitivity of the sensor 119-40 that have been acquired,are transferred from the control computing mechanism 144 to thecomputing mechanism 139, so as to be the spectral sensitivity 140-1,140-2, . . . , and 140-3 to be newly stored into the computing mechanism139, respectively. With this arrangement, after the calibration of thespectral apparatus 100 is performed, the computing mechanism 139 canacquire the spectral characteristics with the first signal, the secondsignal, . . . , and the fortieth signal and the spectral sensitivity ofthe sensor 119-1, the spectral sensitivity of the sensor 119-2, . . . ,and the spectral sensitivity of the sensor 119-40 that have been newlyacquired.

The control computing mechanism 144 is a computer, and operates inaccordance with an installed program. The entirety or part of processingto be performed by the control computing mechanism 144, may be performedby an electronic circuit accompanying no program. The entirety or partof processing to be performed by the control computing mechanism 144,may be manually performed. The control computing mechanism 144 may bebuilt in the spectral apparatus 100.

7. Wavelengths of Emission-Line Components

The light to be measured for emission-line calibration includesemission-line components 145-1, 145-2, 145-3, 145-4, 145-5, and 145-6,as illustrated in FIG. 5. The wavelengths of the emission-linecomponents 145-1, 145-2, 145-3, 145-4, 145-5, and 145-6 are 404.55 nm,435.84 nm, 508.58 nm, 546.07 nm, 578 nm, and 647.85 nm, respectively.The emission-line components 145-1, 145-2, 145-3, 145-4, 145-5, and145-6 are used for the calibration of the spectral apparatus 100.

Emission-line components other than the emission-line components 145-1,145-2, 145-3, 145-4, 145-5, and 145-6, may be used for the calibrationof the spectral apparatus 100. Not more than five emission-linecomponents or not less than seven emission-line components may be usedfor the calibration of the spectral apparatus 100. Light emitted from anemission-line light source other than the HgCd lamp 143, may be used asthe light to be measured for emission-line calibration. Calibration notbeing emission-line calibration may be performed and light emitted froma light source not being an emission-line light source may be used asthe light to be measured for calibration.

8. Models Used in Calibration of Spectral Apparatus

A first model, a second model, or a third model is used in thecalibration of the spectral apparatus 100. The first model and thesecond model are exemplary references.

9. Indicator Indicating Spectral Sensitivity of Each Sensor

The sensors 119-1, 119-2, . . . , and 119-40 are identified with aposition i in each of the first model, the second model, and the thirdmodel. The position i takes any of mutually different 40 values i₁, i₂,. . . , and i₄₀. The sensors 119-1, 119-2, . . . , and 119-40 may beidentified by an indicator other than the position. For example, thesensors 119-1, 119-2, . . . , and 119-40 may be identified by the centerwavelength in reference spectral sensitivity or pixel number.

In each of the first model, the second model, and the third model, thespectral sensitivity of the sensor at the position i is indicated withthe center wavelength λ_(G)(i) and the full width at half maximumFWHM(i) in the spectral sensitivity of the sensor at the position i. Thecenter wavelength λ_(G)(i) and the full width at half maximum FWHM(i)each are a function of the position i.

The spectral sensitivity of the sensor at the position i is favorablyapproximated by a Gaussian function having an independent variable inwavelength and a dependent variable in sensitivity. The shape of theGaussian function is determined with the center wavelength and the fullwidth at half maximum. Therefore, the center wavelength λ_(G)(i) and thefull width at half maximum FWHM(i) are preferable to an indicatorindicating the spectral sensitivity of the sensor at the position i.Note that the indicator indicating the spectral sensitivity of thesensor at the position i may be changed. The indicator indicating thespectral sensitivity of the sensor at the position i, is allowed to haveone variable or not less than three variables.

10. First Model (Exemplary Reference)

In a case where the first model is used, Expression (1) in which thecenter wavelength λ_(G)(i) is expressed by an n-th order function of theposition i, is created.[Mathematical Formula 1]λ_(G)(i)=f ₁(i)=a _(n) ·i ^(n) + . . . +a ₀  (1)

Expression (2) in which the full width at half maximum FWHM(i) isexpressed by an m-th order function of the position i, is created.[Mathematical Formula 2]FWHM(i)=f ₂(i)=b _(m) ·i ^(m) + . . . +b ₀  (2)

The first model includes Expression (1) and Expression (2). Coefficientsa_(n) to a₀ are explanatory variables, and are calibration parametersthat determine the spectral sensitivity of the sensor at the position i.Coefficients b_(n) to b₀ are explanatory variables, and are calibrationparameters that determine the spectral sensitivity of the sensor at theposition i.

In a case where the calibration of the spectral apparatus 100 isperformed with the first model, the coefficients a_(n) to a₀ and thecoefficients b_(n) to b₀ are acquired to adapt a spectral sensitivityset for a set of two sensors or more in the spectral sensitivity of thesensor indicated with the center wavelength λ_(G)(i) and the full widthat half maximum FWHM(i), to a signal set for the set of two sensors ormore in the signal output from the sensor.

Subsequently, the center wavelength λ_(G)(i) is acquired with Expression(1) that has been created and the coefficients a_(n) to a₀ that havebeen acquired, and the full width at half maximum FWHM(i) is acquiredwith Expression (2) that has been created and the coefficients b_(n) tob₀ that have been acquired.

Subsequently, the spectral sensitivity of the sensor at the position iis acquired with the center wavelength λ_(G)(i) and the full width athalf maximum FWHM(i) that have been acquired. The spectral sensitivityof the sensor at the position i is the spectral sensitivity indicatedwith the center wavelength λ_(G)(i) and the full width at half maximumFWHM(i) that have been acquired.

In the case where the calibration of the spectral apparatus 100 isperformed with the first model, the spectral sensitivity is preciselyacquired in wavelength regions close to any of the wavelengthsλ_(HgCd)(1), λ_(HgCd)(2), . . . , and λ_(HgCd)(K₀) of the emission-linecomponents, but the spectral sensitivity is not precisely acquired inthe other wavelength regions. Particularly, the spectral sensitivity isnot precisely acquired in wavelength regions on the shortest wavelengthside and on the longest wavelength side.

11. Second Model (Exemplary Reference)

The second model is provided to solve the problem in the first model.

In a case where the second model is used, an ideal spectral apparatus100 including, for example, the arrangements, shapes, and sizes of theslit plate 115, the concave diffraction grating 116, and the lineararray sensor 112 as designed, is assumed, and the reference spectralsensitivity of the sensor at the position i included in the spectralapparatus 100 that has been assumed, is acquired by optical simulation.The reference spectral sensitivity of the sensor at the position i is afunction of the position i.

Subsequently, the center wavelength λ_(G0)(i) and the full width at halfmaximum FWHM₀(i) in the reference spectral sensitivity of the sensor atthe position i are acquired.

The center wavelength λ_(G0)(i) and the full width at half maximumFWHM₀(i) are preferable to an indicator indicating the referencespectral sensitivity of the sensor at the position i.

Subsequently, Expression (3) in which the deviation Δλ_(G)(i) of thecenter wavelength λ_(G)(i) from the center wavelength λ_(G0)(i) isexpressed by a linear function of the position i, is created.[Mathematical Formula 3]Δλ_(G)(i)=λ_(G)(i)−λ_(G0)(i)=a ₁ ·i+a ₀  (3)

Expression (4) in which the ratio ratio(i) of the full width at halfmaximum FWHM(i) to the full width at half maximum FWHM₀(i) is expressedby a linear function of the position i, is created.

$\begin{matrix}\left\lbrack {{Mathematical}\mspace{14mu}{Formula}\mspace{14mu} 4} \right\rbrack & \; \\{{{ratio}(i)} = {\frac{{FWHM}(i)}{{FWHM}_{0}(i)} = {{b_{1} \cdot i} + b_{0}}}} & (4)\end{matrix}$

The second model includes Expression (3) and Expression (4).Coefficients a₁ and a₀ are explanatory variables, and are calibrationparameters that determine the spectral sensitivity of the sensor at theposition i. Coefficients b₁ and b₀ are explanatory variables, and arecalibration parameters that determine the spectral sensitivity of thesensor at the position i.

The center wavelength λ_(G)(i) is acquired with Expression (5) derivedfrom Expression (3).[Mathematical Formula 5]λ_(G)(i)=λ_(G0)(i)+a ₁ +a ₀  (5)

In a case where the calibration of the spectral apparatus 100 isperformed with the second model, the coefficients a₁ and a₀ and thecoefficients b₁ and b₀ are acquired to adapt a spectral sensitivity setfor a set of two sensors or more in the spectral sensitivity of thesensor indicated with the center wavelength λ_(G)(i) and the full widthat half maximum FWHM(i), to a signal set for the set of two sensors ormore in the signal output from the sensor.

Subsequently, the deviation Δλ_(G)(i) is acquired with Expression (3)that has been created and the coefficients a₁ and a₀ that have beenacquired, and the ratio ratio(i) is acquired with Expression (4) thathas been created and the coefficients b₁ and b₀ that have been acquired.

Subsequently, the spectral sensitivity of the sensor at the position iis acquired with the deviation Δλ_(G)(i) and the ratio ratio(i) thathave been acquired.

The schematic graphical representation of FIG. 6 illustrates therelationship between the reference spectral sensitivity of the sensor atthe position i and the spectral sensitivity of the sensor at theposition i.

As illustrated in FIG. 6, the spectral sensitivity of the sensor at theposition i is acquired by enlarging the reference spectral sensitivityof the sensor at the position i ratio(i) times along the wavelengthaxis, with the center wavelength λ_(G0)(i) centered, and moving thereference spectral sensitivity that has been enlarged, by Δλ_(G)(i)along the wavelength axis.

The graph of FIG. 7 illustrates measured data indicating therelationship between the center wavelength λ_(G0) in the referencespectral sensitivity and the deviation Δλ_(G) in the spectral apparatus100 in practice. The graph of FIG. 8 illustrates measured dataindicating the relationship between the center wavelength λ_(G0) in thereference spectral sensitivity and the ratio in the spectral apparatus100 in practice.

In the spectral apparatus 100 in practice, as illustrated in FIG. 7, itcannot necessarily be said that the deviation Δλ_(G) is a linearfunction of the center wavelength λ_(G0) in the reference spectralsensitivity, and, as illustrated in FIG. 8, it cannot necessarily besaid that the ratio is a linear function of the center wavelengthλ_(G0). In such cases, Expression (3) and Expression (4) each are notfavorable approximation. Therefore, in the case where the calibration ofthe spectral apparatus 100 is performed with the second model, thespectral sensitivity is not necessarily acquired precisely.

In a case where the deviation Δλ_(G)(i) is expressed by a higher orderfunction being a quadratic function of the position i or more and theratio ratio(i) is expressed by a higher order function being a quadraticfunction of the position i or more in order to solve the problem, theadaptability of the spectral sensitivity set to the signal set greatlyvaries when the coefficients a₁, a₀, b₁, and b₀ vary. Therefore, thecoefficients a₁, a₀, b₁, and b₀ are not necessarily acquiredappropriately.

12. Third Model

12.1 Creation of Third Model

The third model is provided in order to solve the problem in the secondmodel.

The flowchart of FIG. 9 illustrates a procedure of creating the thirdmodel.

Even in a case where the calibration of the spectral apparatus 100 isperformed for each of a plurality of spectral apparatuses, the procedureindicated at steps S1 and S2 of FIG. 9 is not required to be performedfor each of the plurality of spectral apparatuses, and thus theprocedure at steps S1 and S2 illustrated in FIG. 9 is required at leastto be performed previously once. Note that, the procedure illustrated inFIG. 9 may be performed again and the third model may be recreatedduring the performance of the calibration of the spectral apparatus 100for each of the plurality of spectral apparatuses.

In a case where the third model is created, at step S, an ideal spectralapparatus 100 including, for example, the arrangements, shapes, andsizes of the slit plate 115, the concave diffraction grating 116, andthe linear array sensor 112 as designed, is assumed, and the referencespectral sensitivity of the sensor at the position i included in thespectral apparatus 100 that has been assumed, is acquired by opticalsimulation. The reference spectral sensitivity of the sensor at theposition i is a function of position i. Therefore, at step S1, thereference spectral sensitivity of the sensor is acquired for each of thesensor at the position i₁, the sensor at the position i₂, . . . , andthe sensor at the position i₄₀.

Subsequently, at step S2, the center wavelength λ_(G0)(i) and the fullwidth at half maximum FWHM₀(i) in the reference spectral sensitivity ofthe sensor at the position i are acquired. The center wavelengthλ_(G0)(i) and the full width at half maximum FWHM₀(i) each are afunction of the position i. Therefore, at step S2, the center wavelengthand the full width at half maximum in the reference spectral sensitivityof the sensor are acquired for each of the sensor at the position i₁,the sensor at the position i₂, . . . , and the sensor at the positioni₄₀.

The center wavelength λ_(G0)(i) and the full width at half maximumFWHM₀(i) are preferable to an indicator indicating the referencespectral sensitivity of the sensor at the position i.

Subsequently, at step S3, Expression (6) in which the deviation of thewavelength λ_(G)(i) from the center wavelength λ_(G0)(i) is expressed bylinear functions of arrangement errors a₁, a₂, and a₃ of the lineararray sensor 112, a manufacturing error a₄ in the width of the slit 122,and an arrangement error as of the concave diffraction grating 116, iscreated. Expression (7) in which the deviation of the full width at halfmaximum FWHM(i) from the full width at half maximum FWHM₀(i) isexpressed by linear functions of the arrangement errors a₁, a₂, and a₃of the linear array sensor 112, the manufacturing error a₄ in the widthof the slit 122, and the arrangement error as of the concave diffractiongrating 116, is created. The center wavelength λ_(G0)(i) and the fullwidth at half maximum FWHM₀(i) each are a function of the position i.The errors a₁, a₂, a₃, a₄, and as are included in an indicatorindicating a mechanical error in the spectral apparatus 100. Therefore,at step S3, the third model in which the deviation of the indicatorindicating the spectral sensitivity of the sensor from the indicatorindicating the reference spectral sensitivity of the sensor is expressedby a linear function of the indicator indicating the mechanical error inthe spectral apparatus 100, is created for each of the sensor at theposition i₁, the sensor at the position i₂, . . . , and the sensor atthe position i₄₀.[Mathematical Formula 6]λ_(G)(i)=λ_(G0)(i)+a ₁·δλ_(G1)(i)+a ₂·δλ_(G2)(i)+a ₃·δλ_(G3)(i)+a₄·δλ_(G4)(i)+a ₅·δλ_(G5)(i)  (6)[Mathematical Formula 7]FWHM(i)=FWHM₀(i)+a ₁·δFWHM₁(i)+a ₂·δFWHM₂(i)+a ₃·δFWHM₃(i)+a₄·δFWHM₄(i)+a ₅·δFWHM₅(i)  (7)

The third model includes Expression (6) and Expression (7). The errorsa₁, a₂, a₃, a₄, and as are explanatory variables, and are calibrationparameters that determine the spectral sensitivity of the sensor at theposition i.

The arrangement error a₁ of the linear array sensor 112 is anarrangement error of the linear array sensor 112 in the direction 128 inwhich the sensors 119-1, 119-2, . . . , and 119-40 are arranged. Thearrangement error a₂ of the linear array sensor 112 is an arrangementerror of the linear array sensor in a direction 129 in which the opticalaxis 135 extends. The arrangement error a₃ of the linear array sensor112 is a turn error in the circumferential direction 130 of an axis 130c of the sensor orthogonal to the direction 128 and the optical axis135. The manufacturing error a₄ in the width of the slit 122 is amanufacturing error in the width of the slit 122 in a direction 131parallel to a principal section 146 of the concave diffraction grating116 and perpendicular to the optical axis 134. The arrangement error a₅is a turn error in the circumferential direction 132 of an axis 132 c ofthe concave diffraction grating 116. The principal section 146 of theconcave diffraction grating 116 is a plane perpendicular to ruled linesformed on the diffraction surface 124.

Since the errors a₁, a₂, a₃, a₄, and a₅ are expected to be small,expressing the deviation of the center wavelength λ_(G)(i) from thewavelength λ_(G0)(i) by the linear functions of the errors a₁, a₂, a₃,a₄, and a₅ is favorable approximation, and expressing the deviation ofthe full width at half maximum FWHM(i) from the full width at halfmaximum FWHM₀(i) by the linear functions of the errors a₁, a₂, a₃, a₄,and a₅ is also favorable approximation. Therefore, in a case where thecalibration of the spectral apparatus 100 is performed with the thirdmodel, the spectral sensitivity is precisely acquired.

The errors a₁, a₂, a₃, a₄, and a₅ have large influence on the centerwavelength λ_(G)(i) or the full width at half maximum FWHM(i), and havelarge influence on the spectral sensitivity of the sensor at theposition i. Therefore, the errors a₁, a₂, a₃, a₄, and a₅ are preferableto the indicator indicating the mechanical error in the spectralapparatus 100. Note that the indicator indicating the mechanical errorin the spectral apparatus 100 may be changed. The number of theexplanatory variables or the calibration parameters included in theindicator indicating the mechanical error in the spectral apparatus 100,is limited not more than the number of the emission-line componentsincluded in the light to be measured for emission-line calibration, butmay be not more than four or not less than six.

In the linear functions of the errors a₁, a₂, a₃, a₄, and a₅ included inExpression (6), the errors a₁, a₂, a₃, a₄, and a₅ are multiplied bycoefficients δλ_(G1)(i), δλ_(G2)(i), δλ_(G3)(i), δλ_(G4)(i), andδλ_(G5)(i), respectively. The coefficients δλ_(G1)(i), δλ_(G2)(i),δλ_(G3)(i), δλ_(G4)(i), and δλ_(G5)(i) are the deviated amounts of thecenter wavelength λ_(G)(i) in a case where the errors a₁, a₂, a₃, a₄,and a₅ occur in unit amounts, respectively, and are acquired by opticalsimulation. The coefficients δλ_(G1)(i), δλ_(G2)(i), δλ_(G3)(i),δλ_(G4)(i), and δλ_(G5)(i) each are a function of the position i.

In the linear functions of the errors a₁, a₂, a₃, a₄, and as included inExpression (7), the errors a₁, a₂, a₃, a₄, and as are multiplied bycoefficients δFWHM₁(i), δFWHM₂(i), δFWHM₃(i), δFWHM₄(i), and δFWHM₅(i),respectively (added and subtracted). The coefficients δFWHM₁(i),δFWHM₂(i), δFWHM₃(i), δFWHM₄(i), and δFWHM(i) are the deviated amountsof the full width at half maximum FWHM(i) in a case where the errors a₁,a₂, a₃, a₄, and as occur in unit amounts, respectively, and are acquiredby optical simulation. The coefficients δFWHM₁(i), δFWHM₂(i), δFWHM₃(i),δFWHM₄(i), and δFWHM(i) each are a function of the position i.

Channels Ch in each of FIGS. 10 to 19 represent the identificationnumbers of the sensors 119-1, 119-2, . . . , and 119-40.

Reference numeral 149 of FIG. 10 represents the deviated amount δλ_(G)of the center wavelength in the spectral sensitivity of each channel Chin a case where the arrangement error a₁ is deviated from 0 by +1 unit,and reference numeral 150 represents the deviated amount δλ_(G) of thecenter wavelength in the spectral sensitivity of each channel Ch in acase where the arrangement error a₁ is deviated from 0 by −1 unit.Relationships 149 and 150 are acquired by optical simulation. Thecoefficient δλ_(G1)(i) is, for example, acquired with relationships 149and 150.

Reference numeral 153 of FIG. 11 represents the deviated amount δFWHM ofthe full width at half maximum in the spectral sensitivity of thechannel Ch of each of the sensors in a case where the arrangement errora₁ is deviated from 0 by +1 unit, and reference numeral 154 representsthe deviated amount δFWHM of the full width at half maximum in thespectral sensitivity of the channel Ch of each of the sensors in a casewhere the arrangement error a₁ is deviated from 0 by −1 unit.Relationships 153 and 154 are acquired by optical simulation. Thecoefficient δFWHM₁(i) is, for example, acquired with relationships 153and 154.

Reference numeral 157 of FIG. 12 represents the deviated amount δλ_(G)of the center wavelength in the spectral sensitivity of each channel Chin a case where the arrangement error a₂ is deviated from 0 by +1 unit,and reference numeral 158 represents the deviated amount δλ_(G) of thecenter wavelength in the spectral sensitivity of each channel Ch in acase where the arrangement error a₂ is deviated from 0 by −1 unit.Relationships 157 and 158 are acquired by optical simulation. Thecoefficient δλ_(G2)(i) is, for example, acquired with relationships 157and 158.

Reference numeral 161 of FIG. 13 represents the deviated amount δFWHM ofthe full width at half maximum in the spectral sensitivity of thechannel Ch of each of the sensors in a case where the arrangement errora₂ is deviated from 0 by +1 unit, and reference numeral 162 representsthe deviated amount δFWHM of the full width at half maximum in thespectral sensitivity of the channel Ch of each of the sensors in a casewhere the arrangement errors a₂ is deviated from 0 by −1 unit.Relationships 161 and 162 are acquired by optical simulation. Thecoefficient δFWHM₂(i) is, for example, acquired with relationships 161and 162.

Reference numeral 165 of FIG. 14 represents the deviated amount δλ_(G)of the center wavelength in the spectral sensitivity of each channel Chin a case where the arrangement error a₃ is deviated from 0 by +1 unit,and reference numeral 166 represents the deviated amount δλ_(G) of thecenter wavelength in the spectral sensitivity of each channel Ch in acase where the arrangement error a₃ is deviated from 0 by −1 unit.Relationships 165 and 166 are acquired by optical simulation. Thecoefficient δλ_(G3)(i) is, for example, acquired with relationships 165and 166.

Reference numeral 169 of FIG. 15 represents the deviated amount δFWHM ofthe full width at half maximum in the spectral sensitivity of thechannel Ch of each of the sensors in a case where the arrangement errora₃ is deviated from 0 by +1 unit, and reference numeral 170 representsthe deviated amount δFWHM of the full width at half maximum in thespectral sensitivity of the channel Ch of each of the sensors in a casewhere the arrangement errors a₃ is deviated from 0 by −1 unit.Relationships 169 and 170 are acquired by optical simulation. Thecoefficient δFWHM₃(i) is, for example, acquired with relationships 169and 170.

Reference numeral 173 of FIG. 16 represents the deviated amount δλ_(G)of the center wavelength in the spectral sensitivity of each channel Chin a case where the manufacturing error a₄ is deviated from 0 by +1unit, and reference numeral 174 represents the deviated amount δλ_(G) ofthe center wavelength in the spectral sensitivity of each channel Ch ina case where the manufacturing error a₄ is deviated from 0 by −1 unit.Relationships 173 and 174 are acquired by optical simulation. Thecoefficient δλ_(G4)(i) is, for example, acquired with relationships 173and 174.

Reference numeral 177 of FIG. 17 represents the channel Ch of thedeviated amount δFWHM of the full width at half maximum in the spectralsensitivity of the channel Ch of each of the sensors in a case where themanufacturing error a₄ is deviated from 0 by +1 unit, and referencenumeral 178 represents the deviated amount δFWHM of the full width athalf maximum in the spectral sensitivity of the channel Ch of each ofthe sensors in a case where the manufacturing errors a₄ is deviated from0 by −1 unit. Relationships 177 and 178 are acquired by opticalsimulation. The coefficient δFWHM₄(i) is, for example, acquired withrelationships 177 and 178.

Reference numeral 181 of FIG. 18 represents the deviated amount δλ_(G)of the center wavelength in the spectral sensitivity of each channel Chin a case where the arrangement error as is deviated from 0 by +1 unit,and reference numeral 182 represents the deviated amount δλ_(G) of thecenter wavelength in the spectral sensitivity of each channel Ch in acase where the arrangement error as is deviated from 0 by −1 unit.Relationships 181 and 182 are acquired by optical simulation. Thecoefficient δλ_(G5)(i) is, for example, acquired with relationships 181and 182.

Reference numeral 185 of FIG. 19 represents the deviated amount δFWHM ofthe full width at half maximum in the spectral sensitivity of thechannel Ch of each of the sensors in a case where the arrangement erroras is deviated from 0 by +1 unit, and reference numeral 186 representsthe deviated amount δFWHM of the full width at half maximum in thespectral sensitivity of the channel Ch of each of the sensors in a casewhere the arrangement errors as is deviated from 0 by −1 unit.Relationships 185 and 186 are acquired by optical simulation. Thecoefficient δFWHM₅(i) is, for example, acquired with relationships 185and 186.

12.2 Calibration of Spectral Apparatus with Third Model

The flowchart of FIG. 20 illustrates a procedure of calibrating thespectral apparatus with the third model.

In a case where the calibration of the spectral apparatus 100 isperformed for each of the plurality of spectral apparatuses, theprocedure illustrated in FIG. 20 is performed for each of the pluralityof spectral apparatuses.

In a case where the calibration of the spectral apparatus 100 isperformed with the third model, at step S11, the errors a₁, a₂, a₃, a₄,and as are acquired to adapt the spectral sensitivity of the sensorindicated with the center wavelength λ_(G)(i) and the full width at halfmaximum FWHM(i), to the signal output by the sensor. Adapting thespectral sensitivity to the signal means causing the signal assumed tobe output by the sensor to be close to the signal output by the sensorin practice, in a case where the real spectral sensitivity of the sensorat the position i is the spectral sensitivity of the sensor indicatedwith the center wavelength λ_(G)(i) and the full width at half maximumFWHM(i). A response variable is used for evaluation of the adaptability.At step S11, the indicator indicating the mechanical error in thespectral apparatus 100, is acquired to adapt the spectral sensitivity ofthe sensor indicated by the indicator indicating the spectralsensitivity of the sensor, to the signal output by the sensor.

The outputs of sensors having sensitivity at the wavelengthsλ_(HgCd)(1), λ_(HgCd)(2), . . . , and λ_(HgCd)(K₀) of the emission-linecomponents, from the sensors 119-1, 119-2, . . . , and 119-40, are usedfor the calibration.

Subsequently, at step S12, the deviation of the center wavelengthλ_(G)(i) from the center wavelength λ_(G0)(i) is acquired withExpression (6) that has been created and the errors a₁, a₂, a₃, a₄, anda₅ that have been acquired. The deviation of the full width at halfmaximum FWHM(i) from the full width at half maximum FWHM₀(i) is acquiredwith Expression (7) that has been created and the errors a₁, a₂, a₃, a₄,and a₅ that have been acquired. At step S12, for each of the sensor atthe position i, the sensor at the position i₂, . . . , and the sensor atthe position i₄₀, the deviation of the indicator indicating the spectralsensitivity of the sensor from the indicator indicating the referencespectral sensitivity of the sensor, is acquired with the third modelthat has been created and the indicator indicating the mechanical errorin the spectral apparatus 100, that has been acquired.

Subsequently, at step S13, the spectral sensitivity of the sensor at theposition i is acquired with the reference spectral sensitivity of thesensor at the position i, the deviation of the center wavelengthλ_(G)(i) from the center wavelength λ_(G0)(i), and the deviation of thefull width at half maximum FWHM(i) from the full width at half maximumFWHM₀(i), that have been acquired. With this arrangement, for each ofthe sensor at the position i₁, the sensor at the position i₂, . . . ,and the sensor at the position i₄₀, the spectral sensitivity of thesensor is acquired with the reference spectral sensitivity of the sensorand the deviation of the indicator indicating the spectral sensitivityof the sensor from the indicator indicating the reference spectralsensitivity of the sensor, that have been acquired.

As illustrated in FIG. 6, the spectral sensitivity of the sensor at theposition i is acquired by enlarging the reference spectral sensitivitythe ratio (i) times along the wavelength axis, with the centerwavelength λ_(G0)(i) centered, and moving the reference spectralsensitivity that has been enlarged, by Δλ_(G)(i) along the wavelengthaxis.

The ratio ratio(i) is expressed by Expression (8).

$\begin{matrix}\left\lbrack {{Mathematical}\mspace{14mu}{Formula}\mspace{14mu} 8} \right\rbrack & \; \\{\begin{matrix}{{{ratio}(i)} = \frac{{FWHM}(i)}{{FWHM}_{0}(i)}} \\{= {1 + {\frac{1}{{FWHM}_{0}} \cdot}}} \\{\left( {{{a_{1} \cdot \delta}\mspace{11mu}{{FWHM}_{1}(i)}} + {{a_{2} \cdot \delta}\mspace{11mu}{{FWHM}_{2}(i)}} +} \right.} \\{{{a_{3} \cdot \delta}\mspace{11mu}{{FWHM}_{3}(i)}} + {{a_{4} \cdot \delta}\mspace{11mu}{{FWHM}_{4}(i)}} +} \\\left. {{a_{5} \cdot \delta}\mspace{11mu}{{FWHM}_{5}(i)}} \right)\end{matrix}\quad} & (8)\end{matrix}$

In the case where the calibration of the spectral apparatus 100 isperformed with the third model, since each of the center wavelengthλ_(G0)(i) and the full width at half maximum FWHM₀(i) is expressed bythe linear functions of the errors a₁, a₂, a₃, a₄, and as, theadaptability of the spectral sensitivity to the signal does not varygreatly even in a case where the errors a₁, a₂, a₃, a₄, and as vary.Therefore, the errors a₁, a₂, a₃, a₄, and as are appropriately acquiredand the calibrated spectral sensitivity of the sensor is appropriatelyacquired.

In the case where the calibration of the spectral apparatus 100 isperformed with the third model, since each of the center wavelengthλ_(G)(i) and the full width at half maximum FWHM₀(i) is expressed by theerrors a₁, a₂, a₃, a₄, and as being the explanatory variables in common,the number of explanatory variables reduces in comparison to a casewhere the center wavelength and the full width at half maximum each areexpressed by different explanatory variables, and thus the number ofemission-line components necessary for the calibration of the spectralapparatus 100 reduces.

Furthermore, in the case where the calibration of the spectral apparatus100 is performed with the third model, since the center wavelengthλ_(G)(i) and the full width at half maximum FWHM₀(i) are expressed bythe errors a₁, a₂, a₃, a₄, and as being the explanatory variables incommon, the relationship between the center wavelength λ_(G0)(i) and thefull width at half maximum FWHM₀(i) does not become inappropriate andthus the center wavelength λ_(G0)(i) and the full width at half maximumFWHM₀(i) are appropriately acquired.

13. Procedure of Adapting Spectral Sensitivity to Signal

In the following, the emission-line component having a wavelength ofλ_(HgCd)(k) is defined to be incident over the sensor at a positionI_(k) and the sensor at the position I_(k+1). The center wavelength inthe reference spectral sensitivity of the sensor at the position I_(k+1)is adjacent to the center wavelength in the reference spectralsensitivity of the sensor at the position I_(k). The identificationnumber k of the emission-line component is defined to take any value of1, 2, . . . , and K₀.

The graph of FIG. 21 exemplarily illustrates the spectral sensitivity ofeach sensor and an emission-line spectrum of the light to be measuredfor emission-line calibration. The graph of FIG. 21 is acquired byenlarging a range of 390 nm to 420 nm in wavelength in the graph of FIG.5.

The emission-line component 145-1 being 404.54 nm in wavelength, isincident over the sensor having the spectral sensitivity 141-5 beingapproximately 396 nm in center wavelength and the sensor having thespectral sensitivity 141-6 being approximately 407 nm in centerwavelength. Therefore, the sensor having the spectral sensitivity 141-5and the sensor having the spectral sensitivity 141-6 each havesensitivity to the emission-line component 145-1 as illustrated in FIG.21.

The graph of FIG. 22 illustrates the relationship between the channelsof the sensors and the signals output by the sensors.

The sensor having channel 5 and the sensor having channel 6 each havesensitivity to the emission-line component 145-1; the sensor havingchannel 8 and the sensor having channel 9 each have sensitivity to theemission-line component 145-2; the sensor having channel 15 and thesensor having channel 16 each have sensitivity to the emission-linecomponent 145-3; the sensor having channel 19 and the sensor havingchannel 20 each have sensitivity to the emission-line component 145-4;the sensor having channel 22 and the sensor having channel 23 each havesensitivity to the emission-line component 145-5; and the sensor havingchannel 29 and the sensor having channel 30 each have sensitivity to theemission-line component 145-6. As a result, signals illustrated in FIG.22 are acquired.

In a case where the calibration of the spectral apparatus 100 iscompletely performed with the emission-line component being λ_(HgCd)(1)in wavelength, the emission-line component being λ_(HgCd)(2) inwavelength, . . . , and the emission-line component being λ_(HgCd)(K₀)in wavelength, the spectral sensitivity Response(i, λ) of the sensor atthe position i and the signal value Count(i) acquired byanalog-to-digital converting the signal output by the sensor at theposition i satisfy the relationships indicated by Expressions (9) and(10) for k=1, 2, . . . , and K₀.[Mathematical Formula 9]Response(I _(k),λ_(HgCd)(k))=Count(I _(k))  (9)[Mathematical Formula 10]Response(I _(k+1),λ_(HgCd)(k))=Count(I _(k+1))  (10)

In a case where the sensor at the position I_(k) and the sensor at theposition I_(k+1) are selected, each having sensitivity at the wavelengthλ_(HgCd)(k), Expressions (9) and (10) indicate, for k=1, 2, . . . , andK₀, the sensitivity Response(I_(k), λ_(HgCd)(k)) at the wavelengthλ_(HgCd)(k) in the spectral sensitivity of the sensor at the positionI_(k), agreeing with the signal value Count(I_(k)) indicating the signaloutput by the sensor at the position I_(k), and the sensitivityResponse(I_(k+1), λ_(HgCd)(k)) at the wavelength λ_(HgCd)(k) in thespectral sensitivity of the sensor at the position I_(k+1), agreeingwith the signal value Count(I_(k+1)) indicating the signal output by thesensor at the position I_(k+1), respectively.

Adapting a spectral sensitivity set to a signal set means causing therelationship between the sensitivity Response(I_(k), λ_(HgCd)(k)) andthe signal value Count(I_(k)) to be close to the relationship indicatedby Expression (9) and causing the relationship between the sensitivityResponse(I_(k+1), λ_(HgCd)(k)) and the signal value Count(I_(k+1)) to beclose to the relationship indicated by Expression (10).

Therefore, a first method of adapting the spectral sensitivity to thesignal includes acquiring the errors a₁, a₂, a₃, a₄, and a₅ to minimizea response variable F indicated by Expression (11).

$\begin{matrix}{\mspace{79mu}\left\lbrack {{Mathematical}\mspace{14mu}{Formula}\mspace{14mu} 11} \right\rbrack} & \; \\{{\quad F} = {\sum\limits_{k = 1}^{K_{0}}\left( {\left( {{{Response}\left( {I_{k},{\lambda_{HgCd}(K)}} \right)} - {{Count}\left( I_{k} \right)}} \right)^{2} + \left( {{{Response}\left( {I_{k + 1},{\lambda_{HgCd}\left( {K + 1} \right)}} \right)} - {{Count}\left( I_{k + 1} \right)}} \right)^{2}} \right)}} & (11)\end{matrix}$

The response variable F indicated by Expression (11) totals, for k=1, 2,. . . , and K₀, the sum of the square of the deviation of thesensitivity Response(I_(k), λ_(HgCd)(k)) from the signal valueCount(I_(k)) and the square of the deviation of the sensitivityResponse(I_(k+1), λ_(HgCd)(k)) from the signal value Count(I_(k+1)).

The square of each deviation may be replaced with a different factorhaving an absolute value increasing as the absolute value of thedeviation increases. For example, the square of each deviation may bereplaced with the absolute value of the deviation.

If three sensors having sensitivity to the emission-line wavelengthλ_(HgCd)(k) are present (I_(k−1), I_(k), and I_(k+1)), the deviatedamount between the signal Count(I_(k−1)) and the sensitivityResponse(I_(k−1)) may be added to the response function.

Expression (12) is derived from Expressions (9) and (10).

$\begin{matrix}\left\lbrack {{Mathematical}\mspace{14mu}{Formula}\mspace{14mu} 12} \right\rbrack & \; \\{\frac{{Response}\left( {I_{k + 1},{\lambda_{HgCd}(K)}} \right)}{{Response}\left( {I_{k},{\lambda_{HgCd}(K)}} \right)} = \frac{{Count}\left( I_{k + 1} \right)}{{Count}\left( I_{k} \right)}} & (12)\end{matrix}$

Adapting the spectral sensitivity to the signal means causing therelationship between the sensitivity Response(I_(k), λ_(HgCd)(k)), thesensitivity Response(I_(k+1), λ_(HgCd)(k)), the signal valueCount(I_(k)), and the signal value Count(I_(k+1)), to be close to therelationship indicated by Expression (12).

Therefore, a second method of adapting the spectral sensitivity to thesignal includes acquiring the errors a₁, a₂, a₃, a₄, and a₅ to minimizethe response variable F indicated by Expression (13).

$\begin{matrix}\left\lbrack {{Mathematical}\mspace{14mu}{Formula}\mspace{14mu} 13} \right\rbrack & \; \\{F = {\sum\limits_{k = 1}^{K_{0}}\left( {\frac{{Response}\left( {I_{k + 1},{\lambda_{HgCd}(K)}} \right)}{{Response}\left( {I_{k},{\lambda_{HgCd}(K)}} \right)} = \frac{{Count}\left( I_{k + 1} \right)}{{Count}\left( I_{k} \right)}} \right)^{2}}} & (13)\end{matrix}$

The response function F indicated by Expression (13) totals, for k=1, 2,. . . , and K₀, the deviation of the ratio of the sensitivityResponse(I_(k+1), λ_(HgCd)(k)) to the sensitivity Response(I_(k),λ_(HgCd)(k)) from the ratio of the signal value Count(I_(k+1)) to thesignal value Count(I_(k)).

According to the response function F indicated by Expression (13), thereis no need to normalize the spectral sensitivity Response(i, λ) so thatthe sensitivity Response(I_(k), λ_(HgCd)(k)) and the sensitivityResponse(I_(k+1), λ_(HgCd)(k)) can be compared with the signal valueCount(I_(k)) and the signal value Count(I_(k+1)), respectively.

If the three sensors having the sensitivity to the emission-linewavelength λ_(HgCd)(k) are present (I_(k−1), I_(k), and I_(k+1)), thedeviated amount of the deviated amount between the ratio of the signalCount(I_(k)) and the signal Count(I_(k−1)) and the ratio of thesensitivity Response(I_(k)) and the sensitivity Response(I_(k−1)), maybe added to the response function, in addition to the deviated amountbetween the ratio of the signal Count(I_(k)) and the signalCount(I_(k+1)) and the ratio of the sensitivity Response(I_(k)) and thesensitivity Response(I_(k+1)).

14. Production of Calibrated Spectral Apparatus

In production of a calibrated spectral apparatus, as illustrated in FIG.23, the spectral apparatus 100 is prepared at step S21, and the spectralapparatus 100 that has been prepared, is calibrated at step S22. At stepS21, a business entity that performs the calibration of the spectralapparatus 100, may manufacture the spectral apparatus 100 so that thespectral apparatus 100 is prepared, or the business entity that performsthe calibration of the spectral apparatus 100, may purchase the spectralapparatus 100 from a different business entity so that the spectralapparatus 100 is prepared.

REFERENCE SIGNS LIST

-   -   100 spectral apparatus    -   111 optical system    -   112 linear array sensor    -   115 slit plate    -   116 concave diffraction grating    -   119-1, 119-2, . . . , 119-40 sensors    -   122 slit    -   143 HgCd lamp

The invention claimed is:
 1. A method of creating a model used forcalibration of a spectral apparatus, the spectral apparatus including:an optical system that converts light to be measured into a spectrum;and a light-receiving sensor including a plurality of sensors thatoutputs a plurality of signals, the plurality of sensors includingsensors that output signals indicating respective energy amounts of aplurality of wavelength components included in the spectrum; the methodcreating the model used for the calibration with the model in which alinear function of an indicator indicating a mechanical error in thespectral apparatus, expresses deviation of an indicator indicatingspectral sensitivity of the sensor from the indicator indicating thereference spectral sensitivity of the sensor, wherein the methodcomprises: a step a) of acquiring, for each of the plurality of sensors,reference spectral sensitivity of the sensor; a step b) of acquiring,for each of the plurality of sensors, an indicator indicating thereference spectral sensitivity of the sensor acquired at the step a);and a step c) of creating, for each of the plurality of sensors, themodel in which the linear function of the indicator indicating themechanical error in the spectral apparatus, expresses deviation of theindicator indicating spectral sensitivity of the sensor from theindicator indicating the reference spectral sensitivity of the sensor,acquired at the step b), wherein the indicator indicating the referencespectral sensitivity of the sensor, acquired at the step b), includes acenter wavelength and a full width at half maximum in the referencespectral sensitivity of the sensor, the indicator indicating thespectral sensitivity of the sensor, included in the model created at thestep c), includes a center wavelength and a full width at half maximumin the spectral sensitivity of the sensor, the step c), for each of theplurality of sensors, includes: creating a first expression in which afirst linear function of the indicator indicating the mechanical errorin the spectral apparatus expresses deviation of the center wavelengthin the spectral sensitivity of the sensor from the center wavelength inthe reference spectral sensitivity of the sensor; and creating a secondexpression in which a second linear function of the indicator indicatingthe mechanical error in the spectral apparatus expresses deviation ofthe full width at half maximum in the spectral sensitivity of the sensorfrom the full width at half maximum in the reference spectralsensitivity of the sensor, and the model created at the step c) includesthe first expression and the second expression.
 2. A method of creatinga model used for calibration of a spectral apparatus, the spectralapparatus including: an optical system that converts light to bemeasured into a spectrum; and a light-receiving sensor including aplurality of sensors that outputs a plurality of signals, the pluralityof sensors including sensors that output signals indicating respectiveenergy amounts of a plurality of wavelength components included in thespectrum; the method creating the model used for the calibration withthe model in which a linear function of an indicator indicating amechanical error in the spectral apparatus, expresses deviation of anindicator indicating spectral sensitivity of the sensor from theindicator indicating the reference spectral sensitivity of the sensor,wherein the method comprises: a step a) of acquiring, for each of theplurality of sensors, reference spectral sensitivity of the sensor; astep b) of acquiring, for each of the plurality of sensors, an indicatorindicating the reference spectral sensitivity of the sensor acquired atthe step a); and a step c) of creating, for each of the plurality ofsensors, the model in which the linear function of the indicatorindicating the mechanical error in the spectral apparatus, expressesdeviation of the indicator indicating spectral sensitivity of the sensorfrom the indicator indicating the reference spectral sensitivity of thesensor, acquired at the step b), wherein the light-receiving sensor hasa light-receiving surface, the plurality of sensors is arranged in afirst direction on the light-receiving surface, the optical system hasan optical axis extending in a second direction, the optical axisleading to the light-receiving surface, and the indicator indicating themechanical error in the spectral apparatus, included in the modelcreated at the step c), includes all or part of an arrangement error ofthe light-receiving sensor in the first direction, an arrangement errorof the light-receiving sensor in the second direction, and anarrangement error of the light-receiving sensor in a turning directionaround an axis orthogonal to the first direction and the optical axisleading to the light-receiving surface.
 3. A method of creating a modelused for calibration of a spectral apparatus, the spectral apparatusincluding: an optical system that converts light to be measured into aspectrum; and a light-receiving sensor including a plurality of sensorsthat outputs a plurality of signals, the plurality of sensors includingsensors that output signals indicating respective energy amounts of aplurality of wavelength components included in the spectrum; the methodcreating the model used for the calibration with the model in which alinear function of an indicator indicating a mechanical error in thespectral apparatus, expresses deviation of an indicator indicatingspectral sensitivity of the sensor from the indicator indicating thereference spectral sensitivity of the sensor, wherein the methodcomprises: a step a) of acquiring, for each of the plurality of sensors,reference spectral sensitivity of the sensor; a step b) of acquiring,for each of the plurality of sensors, an indicator indicating thereference spectral sensitivity of the sensor acquired at the step a);and a step c) of creating, for each of the plurality of sensors, themodel in which the linear function of the indicator indicating themechanical error in the spectral apparatus, expresses deviation of theindicator indicating spectral sensitivity of the sensor from theindicator indicating the reference spectral sensitivity of the sensor,acquired at the step b), wherein the optical system includes: a slitplate including a slit formed; and a diffraction grating having aprincipal section and a diffraction surface, the optical system havingan optical axis leading from the slit to the diffraction surface, andthe indicator indicating the mechanical error in the spectral apparatus,included in the model created at the step c), includes a manufacturingerror in a width of the slit in a direction parallel to the principalsection, the direction being perpendicular to the optical axis leadingfrom the slit to the diffraction surface.
 4. A method of creating amodel used for calibration of a spectral apparatus, the spectralapparatus including: an optical system that converts light to bemeasured into a spectrum; and a light-receiving sensor including aplurality of sensors that outputs a plurality of signals, the pluralityof sensors including sensors that output signals indicating respectiveenergy amounts of a plurality of wavelength components included in thespectrum; the method creating the model used for the calibration withthe model in which a linear function of an indicator indicating amechanical error in the spectral apparatus, expresses deviation of anindicator indicating spectral sensitivity of the sensor from theindicator indicating the reference spectral sensitivity of the sensor,wherein the method comprises: a step a) of acquiring, for each of theplurality of sensors, reference spectral sensitivity of the sensor; astep b) of acquiring, for each of the plurality of sensors, an indicatorindicating the reference spectral sensitivity of the sensor acquired atthe step a); and a step c) of creating, for each of the plurality ofsensors, the model in which the linear function of the indicatorindicating the mechanical error in the spectral apparatus, expressesdeviation of the indicator indicating spectral sensitivity of the sensorfrom the indicator indicating the reference spectral sensitivity of thesensor, acquired at the step b), wherein the optical system includes adiffraction grating having a principal section, and the indicatorindicating the mechanical error in the spectral apparatus, included inthe model created at the step c), includes an arrangement error of thediffraction grating in a turning direction in which the principalsection remains flush.
 5. A method of creating a model used forcalibration of a spectral apparatus, the spectral apparatus including:an optical system that converts light to be measured into a spectrum;and a light-receiving sensor including a plurality of sensors thatoutputs a plurality of signals, the plurality of sensors includingsensors that output signals indicating respective energy amounts of aplurality of wavelength components included in the spectrum; the methodcreating the model used for the calibration with the model in which alinear function of an indicator indicating a mechanical error in thespectral apparatus, expresses deviation of an indicator indicatingspectral sensitivity of the sensor from the indicator indicating thereference spectral sensitivity of the sensor, wherein the methodcomprises: a step a) of acquiring, for each of the plurality of sensors,reference spectral sensitivity of the sensor; a step b) of acquiring,for each of the plurality of sensors, an indicator indicating thereference spectral sensitivity of the sensor acquired at the step a);and a step c) of creating, for each of the plurality of sensors, themodel in which the linear function of the indicator indicating themechanical error in the spectral apparatus, expresses deviation of theindicator indicating spectral sensitivity of the sensor from theindicator indicating the reference spectral sensitivity of the sensor,acquired at the step b), wherein the step b) includes acquiring thereference spectral sensitivity of the sensor by optical simulation, andthe step c) includes: acquiring, by optical simulation, a deviatedamount of the indicator indicating the spectral sensitivity of thesensor to deviation of the indicator indicating the mechanical error inthe spectral apparatus, in a unit amount; and adding and subtracting thedeviated amount per unit error amount, acquired with the indicatorindicating the mechanical error in the spectral apparatus, in the linearfunction of the indicator indicating the mechanical error in thespectral apparatus, included in the model created at the step c).
 6. Amethod of producing a calibrated spectral apparatus, the methodcomprising: a step of preparing a spectral apparatus; and a step ofcalibrating the spectral apparatus with a model created by a method ofcreating a model used for calibration of the spectral apparatus, thespectral apparatus including: an optical system that converts light tobe measured into a spectrum; and a light-receiving sensor including aplurality of sensors that outputs a plurality of signals, the pluralityof sensors including sensors that output signals indicating respectiveenergy amounts of a plurality of wavelength components included in thespectrum; the method creating the model used for the calibration withthe model in which a linear function of an indicator indicating amechanical error in the spectral apparatus, expresses deviation of anindicator indicating spectral sensitivity of the sensor from theindicator indicating the reference spectral sensitivity of the sensor,wherein the method creating the model comprises: a step a) of acquiring,for each of the plurality of sensors, reference spectral sensitivity ofthe sensor; a step b) of acquiring, for each of the plurality ofsensors, an indicator indicating the reference spectral sensitivity ofthe sensor acquired at the step a); and a step c) of creating, for eachof the plurality of sensors, the model in which the linear function ofthe indicator indicating the mechanical error in the spectral apparatus,expresses deviation of the indicator indicating spectral sensitivity ofthe sensor from the indicator indicating the reference spectralsensitivity of the sensor, acquired at the step b), wherein thelight-receiving sensor has a light-receiving surface, the plurality ofsensors is arranged in a first direction on the light-receiving surface,the optical system has an optical axis extending in a second direction,the optical axis leading to the light-receiving surface, and theindicator indicating the mechanical error in the spectral apparatus,included in the model created at the step c), includes all or part of anarrangement error of the light-receiving sensor in the first direction,an arrangement error of the light-receiving sensor in the seconddirection, and an arrangement error of the light-receiving sensor in aturning direction around an axis orthogonal to the first direction andthe optical axis leading to the light-receiving surface.